# Cost-benefit analysis versus benefit-only analysis

Hardly anyone cares about statistics directly. People more often care about decisions they need to make with the help of statistics. This suggests that the statistics and decision-making process should be explicitly integrated. The name for this integrated approach is “decision theory.” Problems in decision theory are set up with the goal of maximizing “utility,” the benefit you expect to get from a decision. Equivalently, problems are set up to minimize expected cost. Cost may be a literal monetary cost, but it could be some other measure of something you want to avoid.

I was at a conference this morning where David Draper gave an excellent talk entitled Bayesian Decision Theory in Biostatistics: the Utility of Utility.  Draper presented an example of selecting variables for a statistical model. But instead of just selecting the most important variables in a purely statistical sense, he factored in the cost of collecting each variable. So if two variables make nearly equal contributions to a model, for example, the procedure would give preference to the variable that is cheaper to collect. In short, Draper recommended a cost-benefit analysis rather than the typical (statistical) benefit-only analysis. Very reasonable.

Why don’t people always take this approach? One reason is that it’s hard to assign utilities to outcomes. Dollar costs are often easy to account for, but it can be much harder to assign values to benefits. For example, you have to ask “Benefit for whom?” In a medical context, do you want to maximize the benefit to patients? Doctors? Insurance companies? Tax payers? Regulators? Statisticians? If you want to maximize some combination of these factors, how do you weight the interests of the various parties?

Assigning utilities is hard work, and you can never make everyone happy. No matter how good of a job you do, someone will criticize you. Nearly everyone agrees in the abstract that considering utilities is the way to go, but in practice it is hardly ever done. Anyone who proposes a way to quantify utility is immediately shot down by people who have a better idea. The net result is that rather than using a reasonable but  imperfect idea of utility, no utility is used at all. Or rather no explicit definition of utility is used. There is usually some implicit idea of utility, chosen for mathematical convenience, and that one wins by default. In general, people much prefer to leave utilities implicit.

In the Q&A after his talk, Draper said something to the effect that the status quo persists for a very good reason: thinking is hard work, and it opens you up to criticism.